#245224

order	512
nilpotency class	3
isoclinism	isoclinic class of #245204
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#3192	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1147 conjugacy classes of subgroups; only the 148 normal subgroups are shown.

normal subgroups only

This group has 1147 conjugacy classes of subgroups — too many to draw. Only its 148 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 48 points:

(1,2)(3,6)(4,7)(5,8)(9,15)(10,12)(11,14)(13,16)(17,18)(19,24)(20,32)(21,26)(22,35)(23,27)(25,33)(28,44)(29,38)(30,43)(31,39)(34,41)(36,42)(37,45)(40,47)(46,48);(6,12)(13,15)(17,19,23,31)(18,24,27,39)(20,28,34,46)(21,30,22,29)(25,40,42,37)(26,43,35,38)(32,44,41,48)(33,47,36,45);(1,3)(2,6)(4,9)(5,10)(7,13)(8,12)(11,16)(14,15)(17,20)(18,25)(19,28)(21,33)(22,36)(23,34)(24,37)(26,41)(27,42)(29,45)(30,47)(31,46)(32,35)(38,48)(39,40)(43,44);(1,4)(2,7)(3,9)(5,11)(6,13)(8,14)(10,16)(12,15)(17,21,23,22)(18,26,27,35)(19,29,31,30)(20,33,34,36)(24,38,39,43)(25,41,42,32)(28,45,46,47)(37,48,40,44)

Polycyclic presentation:

2^9|b^2=i,d^2=i,f^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=ghi,c^b=cg,d^b=di,f^b=fi,e^c=eh,f^c=fi,f^d=fh

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 152420);